Question: Ashley is doing some math exercises on a website called Khon Academy. In Khon Academy, you have to get at least $70\%$ of the problems in an exercise right in order to gain proficiency. So far, Ashley has answered correctly $3$ out of $7$ times. Suppose she answers all of the following $q$ questions correctly and gains proficiency in the exercise. Write an inequality in terms of $q$ that models the situation.
Solution: The strategy We know that in the end, Ashley answered at least $70\%$ of the total number of questions correctly. If we let $C$ denote the number of questions that Ashley answered correctly and $T$ denote the total number of questions, we obtain the inequality $\dfrac{C}{T}\geq0.7$. Now, let's express $C$ and $T$ in terms of $q$. Expressing the total number of problems Ashley answered $3$ out of $7$ questions correctly. She then proceeded to answer $q$ more questions. Therefore, the total number of questions that Ashley answered is $7+q$. Expressing the number of correctly answered questions We know that Ashley got $3$ of her first $7$ problems right. She then answers the next $q$ questions correctly. Therefore the number of correctly answered questions is $3+q$. Putting things together We found that $C=3+q$ and $T=7+q$. Since $\dfrac{C}{T}\geq0.7$, we can substitute and find an inequality in terms of $q$ that models the situation. The answer is: $ \dfrac{3+q}{7+q}\geq 0.7 $